第六章 MATLAB解曲線擬合問題

黑龍江八一農(nóng)墾大學(xué)數(shù)學(xué)建模選修課黑龍江八一農(nóng)墾大學(xué)數(shù)學(xué)建模選修課--#-x=l:20;y=x+3*sin(x);p=polyfit(x,y,6)xi=linspace(l,20,100);z=poyval(p,xi);%多項(xiàng)式求值函數(shù)plot(x,y,，o，,xi,z,，k:，,x,y,，b，)legend('原始數(shù)據(jù)'，'6階曲線')結(jié)果：p=0.0000-0.00210.0505-0.59713.6472-9.729511.3304Ficur?I?-1Ficur?I?-1kA//QQC圖6階曲線例4：再用10階多項(xiàng)式擬合程序：x=1:20;y=x+3*sin(x);p=polyfit(x,y,10)xi=linspace(l,20,100);z=polyval(p,xi);plot(x,y,，o，,xi,z,，k:，,x,y,，b，)legendC原始數(shù)據(jù)，,，10階多項(xiàng)式，)結(jié)果：p=Columns1through7-0.01140.1814-1.806511.2360-0.01140.1814-1.806511.2360Columns8through11-42.086188.5907-92.815540.2671說明：可用不同階的多項(xiàng)式來擬合數(shù)據(jù)，但也不是階數(shù)越高擬合的越好?！?多項(xiàng)式曲線求值函數(shù)polyval()調(diào)用格式：y=polyval(p,x)[y,DELTA]=polyval(p,x,s)說明：y=polyval(p,x)為返回對應(yīng)自變量x在給定系數(shù)P的多項(xiàng)式的值。[y,DELTA]=polyval(p,x,s)使用polyfit函數(shù)的選項(xiàng)輸出s得出誤差估計(jì)Y-DELTA。它假設(shè)polyfit函數(shù)數(shù)據(jù)輸入的誤差是獨(dú)立正態(tài)的，并且方差為常數(shù)。則Y土DELTA將至少包含50%的預(yù)測值?！?多項(xiàng)式曲線擬合的評價(jià)和置信區(qū)間函數(shù)polyconf()調(diào)用格式：[Y,DELTA]=polyconf(p,x,s)[Y,DELTA]=polyconf(p,x,s,alpha)說明：[Y,DELTA]=polyconf(p,x,s)使用polyfit函數(shù)的選項(xiàng)輸出s給出Y的95%置信區(qū)間Y土DELTA。它假設(shè)polyfit函數(shù)數(shù)據(jù)輸入的誤差是獨(dú)立正態(tài)的，并且方差為常數(shù)。1-alpha為置信度。例5：給出上面例1的預(yù)測值及置信度為90%的置信區(qū)間。程序：x=0:.1:1;y=[.3.511.41.61.9.6.4.81.52]n=3;

[p,s]=polyfit(x,y,n)alpha=0.05;[Y,DELTA]二polyconf(p,x,s,alpha)結(jié)果：p=16.7832-25.745910.9802-0.0035s=R:[4x4double]df:7normr:1.1406Y=Columns1through7-0.00350.85381.29701.42661.34341.14800.9413Columns8through110.82380.89631.25942.0140DELTA=Columns1through71.36391.15631.15631.15891.13521.12021.1352Columns8through111.15891.15631.15631.3639§5穩(wěn)健回歸函數(shù):robust()穩(wěn)健回歸是指此回歸方法相對于其他回歸方法而言，受異常值的影響較小。調(diào)用格式：b=robustfit(x,y)[b,stats]=robustfit(x,y)[b,stats]=robustfit(x,y,'wfun',tune,'const')說明：b返回系數(shù)估計(jì)向量；stats返回各種參數(shù)估計(jì)；'wfun'指定一個(gè)加權(quán)函數(shù)；tune為調(diào)協(xié)常數(shù)；'const'的值為'on'(默認(rèn)值)時(shí)添加一個(gè)常數(shù)項(xiàng)；為'off'時(shí)忽略常數(shù)項(xiàng)。例6：演示一個(gè)異常數(shù)據(jù)點(diǎn)如何影響最小二乘擬合值與穩(wěn)健擬合。首先利用函數(shù)y=10-2x加上一些隨機(jī)干擾的項(xiàng)生成數(shù)據(jù)集，然后改變一個(gè)y的值形成異常值。調(diào)用不同的擬合函數(shù),通過圖形觀查影響程度。程序：x=(l:10)';y=10-2*x+randn(10,l);y(10)=0;bls二regress(y,[ones(10,l)x])%線性擬合brob=robustfit(x,y)%穩(wěn)健擬合scatter(x,y)holdonplot(x,bls(l)+bls(2)*x,':')plot(x,brob(l)+brob(2)*x,'r‘)結(jié)果：bls=&4452-1.4784brob=10.2934-2.0006T-運(yùn)“T-運(yùn)“比中I'-iXJLi&U!n-mtMK*.!i?Ur阿到異常值的影響，向異常值偏移?！?向自定義函數(shù)擬合對于給定的數(shù)據(jù)，根據(jù)經(jīng)驗(yàn)擬合為帶有待定常數(shù)的自定義函數(shù)。二、向自定義函數(shù)擬合所用函數(shù)：nlinfit()調(diào)用格式：[beta,r,J]=nlinfit(X,y,'fun',betao)說明：beta返回函數(shù)'fun'中的待定常數(shù)；r表示殘差；J表示雅可比矩陣。X,y為數(shù)據(jù);'fun'自定義函數(shù)；betaO待定常數(shù)初值例7：在化工生產(chǎn)中獲得的氯氣的級分y隨生產(chǎn)時(shí)間x下降，假定在x28時(shí)，y與x之間有如下形式的非線性模型：y二a+(0.49—a)e-b(x-8)現(xiàn)收集了44組數(shù)據(jù)，利用該數(shù)據(jù)通過擬合確定非線性模型中的待定常數(shù)。xyxyxy80.49160.43280.4180.49180.46280.40100.48180.45300.40100.47200.42300.40100.48200.42300.38100.47200.43320.41120.46200.41320.40120.46220.41340.40120.45220.40360.41120.43240.42360.36140.45240.40380.40140.43240.40380.40140.43260.41400.36160.44260.40420.39160.43260.41首先定義非線性函數(shù)的m文件：fff6.mfunctionyy二model(betaO,x)a=beta0(l);b=beta0⑵;yy=a+(0.49-a)*exp(-b*(x-8));程序：x=[8.008.0010.0010.0010.0010.0012.0012.0012.0014.0014.0014.00...16.0016.0016.0018.0018.0020.0020.0020.0020.0022.0022.0024.00...24.0024.0026.0026.0026.0028.0028.0030.0030.0030.0032.0032.00...34.0036.0036.0038.0038.0040.0042.00]';y=[0.490.490.480.470.480.470.460.460.450.430.450.430.430.440.43..0.430.460.420.420.430.410.410.400.420.400.400.410.400.410.41...0.400.400.400.380.410.400.400.410.3